在线性代数中，一个 n-by-n 方阵 A. 如果存在n×n平方矩阵，则称为可逆矩阵 B 以至于 
你在哪里 在里面 ‘表示 n-by-n 身份矩阵。矩阵 B 被称为矩阵的逆 A. . 方阵是可逆的当且仅当其行列式非零。
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![图片[2]-检查矩阵是否可逆-yiteyi-C++库](https://www.yiteyi.com/wp-content/uploads/geeks/geeks_Invertible.png)
例如：
Input : {{1, 2, 3} {4, 5, 6} {7, 8, 9}}Output : NoThe given matrix is NOT InvertibleThe value of Determinant is: 0
我们发现 矩阵的行列式 .然后我们检查行列式值是否为0。如果值为0，则我们输出，不可逆。
C++
// C++ program to find Determinant of a matrix #include <bits/stdc++.h> using namespace std; // Dimension of input square matrix #define N 4 // Function to get cofactor of mat[p][q] in temp[][]. n is current // dimension of mat[][] void getCofactor( int mat[N][N], int temp[N][N], int p, int q, int n) { int i = 0, j = 0; // Looping for each element of the matrix for ( int row = 0; row < n; row++) { for ( int col = 0; col < n; col++) { // Copying into temporary matrix only those element // which are not in given row and column if (row != p && col != q) { temp[i][j++] = mat[row][col]; // Row is filled, so increase row index and // reset col index if (j == n - 1) { j = 0; i++; } } } } } /* Recursive function for finding determinant of matrix. n is current dimension of mat[][]. */ int determinantOfMatrix( int mat[N][N], int n) { int D = 0; // Initialize result // Base case : if matrix contains single element if (n == 1) return mat[0][0]; int temp[N][N]; // To store cofactors int sign = 1; // To store sign multiplier // Iterate for each element of first row for ( int f = 0; f < n; f++) { // Getting Cofactor of mat[0][f] getCofactor(mat, temp, 0, f, n); D += sign * mat[0][f] * determinantOfMatrix(temp, n - 1); // terms are to be added with alternate sign sign = -sign; } return D; } bool isInvertible( int mat[N][N], int n) { if (determinantOfMatrix(mat, N) != 0) return true ; else return false ; } // Driver program to test above functions int main() { /* int mat[N][N] = {{6, 1, 1}, {4, -2, 5}, {2, 8, 7}}; */ int mat[N][N] = { { 1, 0, 2, -1 }, { 3, 0, 0, 5 }, { 2, 1, 4, -3 }, { 1, 0, 5, 0 } }; if (isInvertible(mat, N)) cout << "Yes" ; else cout << "No" ; return 0; } |
JAVA
// Java program to find // Determinant of a matrix class GFG { // Dimension of input square matrix static int N = 4 ; // Function to get cofactor // of mat[p][q] in temp[][]. // n is current dimension // of mat[][] static void getCofactor( int [][]mat, int [][]temp, int p, int q, int n) { int i = 0 , j = 0 ; // Looping for each // element of the matrix for ( int row = 0 ; row < n; row++) { for ( int col = 0 ; col < n; col++) { // Copying into temporary matrix // only those element which are // not in given row and column if (row != p && col != q) { temp[i][j++] = mat[row][col]; // Row is filled, so increase // row index and reset col index if (j == n - 1 ) { j = 0 ; i++; } } } } } /* Recursive function for finding determinant of matrix. n is current dimension of mat[][]. */ static int determinantOfMatrix( int [][]mat, int n) { int D = 0 ; // Initialize result // Base case : if matrix // contains single element if (n == 1 ) return mat[ 0 ][ 0 ]; // To store cofactors int [][]temp = new int [N][N]; // To store sign multiplier int sign = 1 ; // Iterate for each // element of first row for ( int f = 0 ; f < n; f++) { // Getting Cofactor of mat[0][f] getCofactor(mat, temp, 0 , f, n); D += sign * mat[ 0 ][f] * determinantOfMatrix(temp, n - 1 ); // terms are to be added // with alternate sign sign = -sign; } return D; } static boolean isInvertible( int [][]mat, int n) { if (determinantOfMatrix(mat, N) != 0 ) return true ; else return false ; } // Driver Code public static void main(String []args) { int [][]mat = {{ 1 , 0 , 2 , - 1 }, { 3 , 0 , 0 , 5 }, { 2 , 1 , 4 , - 3 }, { 1 , 0 , 5 , 0 }}; if (isInvertible(mat, N)) System.out.println( "Yes" ); else System.out.println( "No" ); } } // This code is contributed // by ChitraNayal |
Python 3
# Function to get cofactor of # mat[p][q] in temp[][]. n is # current dimension of mat[][] def getCofactor(mat, temp, p, q, n): i = 0 j = 0 # Looping for each element # of the matrix for row in range (n): for col in range (n): # Copying into temporary matrix # only those element which are # not in given row and column if (row ! = p and col ! = q) : temp[i][j] = mat[row][col] j + = 1 # Row is filled, so increase # row index and reset col index if (j = = n - 1 ): j = 0 i + = 1 # Recursive function for # finding determinant of matrix. # n is current dimension of mat[][]. def determinantOfMatrix(mat, n): D = 0 # Initialize result # Base case : if matrix # contains single element if (n = = 1 ): return mat[ 0 ][ 0 ] # To store cofactors temp = [[ 0 for x in range (N)] for y in range (N)] sign = 1 # To store sign multiplier # Iterate for each # element of first row for f in range (n): # Getting Cofactor of mat[0][f] getCofactor(mat, temp, 0 , f, n) D + = (sign * mat[ 0 ][f] * determinantOfMatrix(temp, n - 1 )) # terms are to be added # with alternate sign sign = - sign return D def isInvertible(mat, n): if (determinantOfMatrix(mat, N) ! = 0 ): return True else : return False # Driver Code mat = [[ 1 , 0 , 2 , - 1 ], [ 3 , 0 , 0 , 5 ], [ 2 , 1 , 4 , - 3 ], [ 1 , 0 , 5 , 0 ]]; N = 4 if (isInvertible(mat, N)): print ( "Yes" ) else : print ( "No" ) # This code is contributed # by ChitraNayal |
C#
// C# program to find // Determinant of a matrix using System; class GFG { // Dimension of input // square matrix static int N = 4; // Function to get cofactor of // mat[p,q] in temp[,]. n is // current dimension of mat[,] static void getCofactor( int [,] mat, int [,] temp, int p, int q, int n) { int i = 0, j = 0; // Looping for each element // of the matrix for ( int row = 0; row < n; row++) { for ( int col = 0; col < n; col++) { // Copying into temporary matrix // only those element which are // not in given row and column if (row != p && col != q) { temp[i, j++] = mat[row, col]; // Row is filled, so // increase row index and // reset col index if (j == n - 1) { j = 0; i++; } } } } } /* Recursive function for finding determinant of matrix. n is current dimension of mat[,]. */ static int determinantOfMatrix( int [,] mat, int n) { int D = 0; // Initialize result // Base case : if matrix // contains single element if (n == 1) return mat[0, 0]; // To store cofactors int [,] temp = new int [N, N]; int sign = 1; // To store sign multiplier // Iterate for each // element of first row for ( int f = 0; f < n; f++) { // Getting Cofactor of mat[0,f] getCofactor(mat, temp, 0, f, n); D += sign * mat[0, f] * determinantOfMatrix(temp, n - 1); // terms are to be added // with alternate sign sign = -sign; } return D; } static bool isInvertible( int [,] mat, int n) { if (determinantOfMatrix(mat, N) != 0) return true ; else return false ; } // Driver Code public static void Main() { int [,] mat = {{ 1, 0, 2, -1 }, { 3, 0, 0, 5 }, { 2, 1, 4, -3 }, { 1, 0, 5, 0 }}; if (isInvertible(mat, N)) Console.Write( "Yes" ); else Console.Write( "No" ); } } // This code is contributed // by ChitraNayal |
PHP
<?php // PHP program to find Determinant // of a matrix // Dimension of input // square matrix $N = 4; // Function to get cofactor of // $mat[$p][$q] in $temp[][]. // $n is current dimension of $mat[][] function getCofactor(& $mat , & $temp , $p , $q , $n ) { $i = 0; $j = 0; // Looping for each element // of the matrix for ( $row = 0; $row < $n ; $row ++) { for ( $col = 0; $col < $n ; $col ++) { // Copying into temporary matrix // only those element which are // not in given row and column if ( $row != $p && $col != $q ) { $temp [ $i ][ $j ++] = $mat [ $row ][ $col ]; // Row is filled, so // increase row index and // reset col index if ( $j == $n - 1) { $j = 0; $i ++; } } } } } /* Recursive function for finding determinant of matrix. n is current dimension of $mat[][]. */ function determinantOfMatrix(& $mat , $n ) { $D = 0; // Initialize result // Base case : if matrix // contains single element if ( $n == 1) return $mat [0][0]; $temp = array ( array ()); // To store cofactors $sign = 1; // To store sign multiplier // Iterate for each // element of first row for ( $f = 0; $f < $n ; $f ++) { // Getting Cofactor of $mat[0][$f] getCofactor( $mat , $temp , 0, $f , $n ); $D += $sign * $mat [0][ $f ] * determinantOfMatrix( $temp , $n - 1); // terms are to be added // with alternate sign $sign = - $sign ; } return $D ; } function isInvertible(& $mat , $n ) { global $N ; if (determinantOfMatrix( $mat , $N ) != 0) return true; else return false; } // Driver Code $mat = array ( array (1, 0, 2, -1 ), array (3, 0, 0, 5 ), array (2, 1, 4, -3 ), array (1, 0, 5, 0 )); if (isInvertible( $mat , $N )) echo "Yes" ; else echo "No" ; // This code is contributed // by ChitraNayal ?> |
Javascript
<script> // Javascript program to find // Determinant of a matrix // Function to get cofactor // of mat[p][q] in temp[][]. // n is current dimension // of mat[][] function getCofactor(mat,temp,p,q,n) { let i = 0, j = 0; // Looping for each // element of the matrix for (let row = 0; row < n; row++) { for (let col = 0; col < n; col++) { // Copying into temporary matrix // only those element which are // not in given row and column if (row != p && col != q) { temp[i][j++] = mat[row][col]; // Row is filled, so increase // row index and reset col index if (j == n - 1) { j = 0; i++; } } } } } /* Recursive function for finding determinant of matrix. n is current dimension of mat[][]. */ function determinantOfMatrix(mat,n) { let D = 0; // Initialize result // Base case : if matrix // contains single element if (n == 1) return mat[0][0]; // To store cofactors let temp = new Array(N); for (let i=0;i<N;i++) { temp[i]= new Array(N); for (let j=0;j<N;j++) { temp[i][j]=0; } } // To store sign multiplier let sign = 1; // Iterate for each // element of first row for (let f = 0; f < n; f++) { // Getting Cofactor of mat[0][f] getCofactor(mat, temp, 0, f, n); D += sign * mat[0][f] * determinantOfMatrix(temp, n - 1); // terms are to be added // with alternate sign sign = -sign; } return D; } function isInvertible(mat,n) { if (determinantOfMatrix(mat, N) != 0) return true ; else return false ; } // Driver Code let mat = [[ 1, 0, 2, -1 ], [ 3, 0, 0, 5 ], [ 2, 1, 4, -3 ], [ 1, 0, 5, 0 ]]; let N = 4 if (isInvertible(mat, N)) document.write( "Yes" ) else document.write( "No" ) // This code is contributed by rag2127 </script> |
输出：
Yes
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